Date: Thu, 11 May 95 17:57:14 -0400
From: michael reid <mreid@ptc.com >
Subject: Re: more on the slice group

jerry writes

There are other positions with the same symmetry characteristics as
the 4-spot. That is, there are other positions for which the
symmetry group contains sixteen elements. There are only three subgroups
of M containing sixteen elements, and the three subgroups are M conjugate.

these subgroups are the 2-sylow subgroups of M. one of sylow's
theorems states that any two p-sylow subgroups are conjugate.

one of these subgroups is the group of symmetries that preserve
the U-D axis. call this subgroup "P". (this is also the group
of symmetries of the intermediate subgroup of kociemba's algorithm.)

jerry asks about P-symmetric positions. coincidentally, i happened
to investigate these a few weeks back, and here's what i found:
(i calculated by hand, so i'd be grateful for any confirmation.)

there are 128 P-symmetric positions, 4 of which are M-symmetric.
they form a subgroup of the cube group (of course) which is
isomorphic to a direct product of 7 copies of C_2. in particular,
each such position has order 2 (or 1) as a group element. thus,
the answer to jerry's question

Call the 4-spot with all edges flipped t. Then, we certainly have
t'=t. Is this true of all positions whose symmetry group contains
sixteen elements?

is "yes". for what it's worth, this group of 128 positions can be
generated by the seven elements

```superflip
pons asinorum
four spots
slice squared               ( U2 D2 )
eight flip                  ( FB UD RL FB UD RL )
four pluses                 ( R2 F2 R2 U'D F2 R2 F2 UD' )
four swapped corner pairs   ( D' B2 U'D F2 U2 L2 B2 L2 B2 U2 L2 F2 U )
```

however, these positions are not all locally maximal; for instance
U2 D2 is not.

mike